This eminent work focuses on the interplay between the behavior of random walks and discrete structure theory. Wolfgang Woess considers Markov chains whose state space is equipped with the structure of an infinite, locally-finite graph, or of a finitely generated group. He assumes the transition probabilities are adapted to the underlying structure in some way that must be specified precisely in each case. He also explores the impact the particular type of structure has on various aspects of the behavior of the random walk. In addition, the author shows how random walks are useful tools for classifying, or at least describing, the structure of graphs and groups.
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Book Description
Editorial Reviews
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"The organization of the book is well-thought-out...The reviewer has a very high opition of this book" Bulletin of the American Mathematical Society
"a very valuable addition to the literture on this fascinating and important subject." Mathematical Review
"a very valuable addition to the literture on this fascinating and important subject." Mathematical Review
Book Description
The main theme of this book is the interplay between the behaviour of a class of stochastic processes (random walks) and discrete structure theory. From the probabilistic viewpoint, the question is what impact the particular type of structure has on various aspects of the behaviour of the random walk. Vice-versa, random walks may also be seen as useful tools for classifying, or at least describing the structure of graphs and groups. This book will be essential reading for all researchers working in stochastic process and related topics.
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5.0 out of 5 stars
A difficult, but a rewarding book,
By PST "A Reader from Germany" (Eislingen Deutschland) - See all my reviews
This review is from: Random Walks on Infinite Graphs and Groups (Cambridge Tracts in Mathematics) (Paperback)
This book is difficult for self-study - and the author says so in his preface! I felt, the prerequities in probablity theory and in graph theory are relatively modest (Prof. Bollobas' book is much more than adequate!), but a excellent knowledge in group theory will very definitely help. For the last chapter, some knowledge in topology would be advantageous, too.There is no doubt, that Prof. Woess is an expert in this field, and the various methods employed to proof theorems are very interesting, and often "unexpected". If there is a point to critizise, it is the relative lack of examples, showing where the stuff proved might be employed, also a few figures to help visualize certain concepts might have added to the book. The book contains an "average" number of typos, I counted about 40 or so, mostly of a harmless nature. For anybody interested in random walks, with a GOOD knowledge of group theory I can thouroughly recommend this book.
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Front Cover | Front Flap | Table of Contents | First Pages | Index | Back Flap | Back Cover | Surprise Me!
Inside This Book
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First Sentence:
Before embarking on a review of the basic material concerning Markov chains, graphs, groups, etc., let us warm up by considering the classical example. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
finite energy flow, hyperbolic compactification, end compactification, strong isoperimetric inequality, neighbour random walk, rough isometry, minimal harmonic functions, step length distributions, adaptedness conditions, irreducible random walk, rough isometries, finite range random, bounded vertex degrees, induced random walk, asymptotic type, hyperbolic graph, nearest neighbour random, simple random walk, local central limit theorem, infinitely many ends, compact normal subgroup, local limit theorem, locally finite graph, recurrence criterion, bounded geometry Key Phrases - Capitalized Phrases (CAPs): (learn more)
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Front Cover | Front Flap | Table of Contents | First Pages | Index | Back Flap | Back Cover | Surprise Me!
Before embarking on a review of the basic material concerning Markov chains, graphs, groups, etc., let us warm up by considering the classical example. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
finite energy flow, hyperbolic compactification, end compactification, strong isoperimetric inequality, neighbour random walk, rough isometry, minimal harmonic functions, step length distributions, adaptedness conditions, irreducible random walk, rough isometries, finite range random, bounded vertex degrees, induced random walk, asymptotic type, hyperbolic graph, nearest neighbour random, simple random walk, local central limit theorem, infinitely many ends, compact normal subgroup, local limit theorem, locally finite graph, recurrence criterion, bounded geometry Key Phrases - Capitalized Phrases (CAPs): (learn more)
Vadim Kaimanovich New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Front Flap | Table of Contents | First Pages | Index | Back Flap | Back Cover | Surprise Me!
Citations (learn more)
This book cites 32
books:
See all 32 books this book cites
5
books
cite this book:
- Real and Complex Analysis (International Series in Pure and Applied Mathematics) by Walter Rudin in Back Matter
- Local Fields (London Mathematical Society Student Texts) by J. W. S. Cassels in Back Matter
- Polycyclic Groups (Cambridge Tracts in Mathematics) by Daniel Segal in Back Matter
- An Introduction to Orthogonal Polynomials (Mathematics and Its Applications) by T. S. Chihara in Back Matter
- Matrix Analysis by Roger A. Horn in Back Matter
See all 32 books this book cites
- Random Walks and Geometry: Proceedings of a Workshop at the Erwin Schrodinger Institute, Vienna, June 18-July 13, 2001 (De Gruyter Proceedings in Mathematics) by Vadim A. Kaimanovich on 5 pages
- Dynamics and Randomness II (Nonlinear Phenomena and Complex Systems) (v. 2) by Alejandro Maass on page 122
- Topological and Asymptotic Aspects of Group Theory (Contemporary Mathematics) by Michael Mihalik on page 45
- Introduction to Circle Packing: The Theory of Discrete Analytic Functions by Kenneth Stephenson in Back Matter
- God and the Universe by Arthur Gibson in Back Matter
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